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Topic: Chromatic polynomial

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	PlanetMath: chromatic number
		The chromatic number of a graph is the minimum number of colours required to colour it.
		In general, however, finding the chromatic number of a large graph (and, similarly, an optimal colouring) is a very difficult (NP-hard) problem.
		This is version 5 of chromatic number, born on 2002-02-03, modified 2004-04-13.
	planetmath.org /encyclopedia/ChromaticNumber.html (143 words)

	Polynomial - Wikipedia, the free encyclopedia
		The derivative of a polynomial is a polynomial
		The integral of a polynomial is a polynomial
		In knot theory the Alexander polynomial, the Jones polynomial, and the HOMFLY polynomial are important knot invariants.
	en.wikipedia.org /wiki/Polynomial (2691 words)

	More on Polynomial
		Because of their simple structure, polynomials are very easy to evaluate, and are used extensively in numerical analysis for polynomial interpolation or to numerically integrate more complex functions.
		To every polynomial f in R[X], one can associate a polynomial function with domain and range equal to R. One obtains the value of this function for a given argument r by everywhere replacing the symbol X in f's expression by r.
		The reason that algebraists have to distinguish between polynomials and polynomial functions is that over some rings R (for instance, over finite fields), two different polynomials may give rise to the same polynomial function.
	www.artilifes.com /polynomial.htm (1982 words)

	PlanetMath: chromatic polynomial
		Compared to the chromatic polynomial, the Tutte contains more information about the matroid.
		Cross-references: variables, function, matroid, chromatic number, span, subsets, sum, Moebius inversion, formula, sign, connected components, integer, argument, induction, polynomial, polynomial function, adjacent, mappings, natural number, edges, multiple, loops, finite, vertices, graph theory, graph
		This is version 5 of chromatic polynomial, born on 2003-09-29, modified 2003-10-19.
	planetmath.org /encyclopedia/ChromaticPolynomial.html (232 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Abstract: We introduce a homology theory for embedded graphs whose graded Euler characteristic is the chromatic polynomial, and whose Poincar\'{e} polynomial is invariant on different planar embeddings of the same graph.
		Both the chromatic polynomial and the Jones polynomial are determined by the Tutte polynomial on the (signed) graphs, or equivalently by the $+-J$ Potts partition function.
		We extend this to the homology: we categorify the Bollob\'{a}s-Riordan topological Tutte polynomial of signed embedded graphs so that the homology of the chromatic polynomial can be recovered by a projection and the Khovanov homology theory of the Jones polynomial can be recovered using a 'universal coefficients theorem'.
	kam.mff.cuni.cz /~iain/abstracts/chromcatabs.html (163 words)

	Edward J. Farrell - Mathematician of the African Diaspora
		Farrell, E. On the derivative of the chromatic polynomial.
		[88] 84b:05082 Farrell, E. On a class of polynomials associated with the subgraphs of a graph and its application to chromatic and dichromatic polynomials.
		Farrell, E. On a class of polynomials obtained from the circuits in a graph and its application to characteristic polynomials of graphs.
	www.math.buffalo.edu /mad/PEEPS/farrell_edwardj.html (1595 words)

	Graph Minors
		The chromatic number of a graph G is the minimum integer k such that G has a k-colouring.
		The edge chromatic number of a graph G is the minimum integer k such that G has a k-edge-colouring.
		The flow polynomial, the chromatic polynomial, the number of spanning trees, and many other peices of information can be deduced from the Tutte polynomial (if one knows the number of vertices).
	www.math.fau.edu /locke/Graphmin.htm (898 words)

	Chromatic Polynomial -- from Wolfram MathWorld
		211), and calculating the chromatic polynomial of a graph is at least an NP-complete problem (Skiena 1990, pp.
		Tutte (1970) showed that the chromatic polynomial of a planar triangulation of a sphere possess a root close to
		The chromatic number of a graph gives the smallest number of colors with which a graph can be colored, and so is the smallest positive integer
	mathworld.wolfram.com /ChromaticPolynomial.html (352 words)

	Tutte Polynomials in Square Grids (Site not responding. Last check: 2007-11-06)
		Finally, the chromatic polynomial of a (possibly disconnected) graph is the product of the chromatic polynomials of its connected components.
		A generalization of the chromatic polynomial is the Tutte polynomial T(G;x,y) of a graph G [
		Chromatic and Tutte polynomials extend to this setting with the same type of properties.
	algo.inria.fr /seminars/sem99-00/noy2.html (1817 words)

	DMTCS vol 6 no 1 (2003), pp. 69-90 (Site not responding. Last check: 2007-11-06)
		We present a two-variable polynomial, which simultaneously generalizes the chromatic polynomial, the independence polynomial, and the matching polynomial of a graph.
		We establish two general expressions for this new polynomial: one in terms of the broken circuit complex and one in terms of the lattice of forbidden colorings.
		We finally give explicit expressions for the generalized chromatic polynomial of complete graphs, complete bipartite graphs, paths, and cycles, and show that it can be computed in polynomial time for trees and graphs of restricted pathwidth.
	dmtcs.loria.fr /volumes/abstracts/dm060106.abs.html (344 words)

	Kee Teo, Institute of Fundamental Sciences
		Chromatic Polynomials, Shanghai (1994); Visiting Professor, Prince of Songkla University, Hat Yai, Thailand (1999); Visiting Scientist, National University of Singapore (2002); Member, NZ Math.
		I am also interested in the general properties of chromatic polynomials, including the nature of their coefficients and roots.
		"Chromatic Polynomials and Chromaticity of Graphs"(with F.M. Dong and K.M. Koh), 2005.
	ifs.massey.ac.nz /staff/teo.shtml (838 words)

	week22
		This is associated to the spin-1 representation of the quantum group SU(2), and it is a polynomial in a variable q that represents e^h, where h is Planck's constant.
		But in 1966 Barri tabulated a bunch of chromatic polynomials in her thesis, and in 1969 Berman and Tutte noticed that most of them had a root that agreed with G + 1 up to at least 5 decimal places.
		For example, the four nonintegral real roots of the chromatic polynomial of the truncated icosahedron are awfully close to B(5), B(7), B(8) and B(9).
	math.ucr.edu /home/baez/week22.html (2124 words)

	Chromatic Number -- from Wolfram MathWorld
		Calculating the chromatic number of a graph is an NP-complete problem (Skiena 1990, pp.
		The chromatic number of a graph must be greater than or equal to its clique number.
		A graph with chromatic number two is said to be bicolorable, and a graph with chromatic number three is said to be three-colorable.
	mathworld.wolfram.com /ChromaticNumber.html (386 words)

	Graph::chromaticPolynomial
		Graph::chromaticPolynomial(G, x) returns the chromatic polynomial of the graph G, using x as indeterminate.
		Graph::chromaticPolynomial(G, x) returns the chromatic polynomial of the graph G.
		In each of the 20 cases, we are left with 3 vertices that form a complete graph and 3 colors, such that there are 6 colourings.
	www.sciface.com /support/doc/40/en/Graph/chromaticPolynomial.html (183 words)

	No Title
		The ``degree'' of a vertex is the number of edges connected to that vertex.
		Consider the chromatic polynomial P(G,x) of a graph G with n vertices and chromatic number k.
		Compare many tables of coefficients from chromatic polynomials of many types of graphs.
	www.math.mtu.edu /~trolson/challenge2/challenge2.html (489 words)

	Research
		The Tutte polynomial is a two variable polynomial encoding a great deal of information about a graph, including the number of spanning trees, spanning forests, acyclic orientations and spanning subsets.
		Along various curves it specialises to the chromatic polynomial, the reliability polynomial, the Ising and q-state Potts models from statistical physics and the Jones polynomial of an alternating link.
		This polynomial includes a vast range of graph invariants as special cases including the matching polynomial and is equivalent (i.e.
	people.brunel.ac.uk /~mastsdn/research.html (552 words)

	Graph Theory Lecture Notes 6
		For a given graph G, the number of ways of coloring the vertices with x or fewer colors is denoted by P(G, x) and is called the chromatic polynomial of G (in terms of x).
		Theorem: (Whitney, 1932): The powers of the chromatic polynomial are consecutive and the coefficients alternate in sign.
		Thus, the smallest exponent in the chromatic polynomial of G is the sum of the smallest exponents appearing in each of the P(G
	www-math.cudenver.edu /~wcherowi/courses/m4408/gtln6.htm (974 words)

	The Reconstruction Conjecture
		Years later, he heard in a lecture that the characteristic polynomial and chromatic number were not known to be reconstructible.
		The chromatic polynomial, chromatic number, the flow polynomial, and the number of spanning trees are easily deduced from the dichromatic polynomial; and Pouzet had shown that the only missing step in reconstructing the characteristic polynomial was counting the number of hamilton cycles.
		Thus, the Tutte polynomial and the characteristic polynomial together are not enough to characterize a graph.
	www.math.fau.edu /locke/Recon.htm (1083 words)

	sm342 Discrete Math, Computer lab 2 (Site not responding. Last check: 2007-11-06)
		computes the characteristic polynomial of the adjacency matrix of G expressed as a polynomial in x.
		returns the chromatic polynomial of the graph G as a polynomial in lambda.
		The value of this polynomial gives the number of proper vertex-colorings of G using lambda colors.
	web.usna.navy.mil /~wdj/teach/sm342/sm342_computer_lab2.html (836 words)

	Tutte polynomial - Wikipedia, the free encyclopedia
		In mathematics, the Tutte polynomial of a matroid can be regarded as a generalisation of the chromatic polynomial of a graph.
		As an application, the chromatic polynomial of a graph is, up to normalisation, a specialisation of the Tutte polynomial.
		Many other graph invariants related to trees and forests, flows, percolation, reliability, and knot polynomials, are evaluations or specialisations of the Tutte polynomial.
	en.wikipedia.org /wiki/Tutte_polynomial (480 words)

	Graph coloring - Wikipedia, the free encyclopedia
		The chromatic polynomial counts the number of ways a graph can be coloured using no more than a given number of colours.
		The chromatic polynomial is a function P(G,t) that counts the number of t-colorings of G.
		In general, computing the chromatic polynomial is Sharp-P-complete, so it is unlikely that a polynomial time algorithm for all graphs will be found.
	en.wikipedia.org /wiki/Chromatic_number (1580 words)

	A New Two-Variable Generalization of the Chromatic Polynomial - Dohmen, Tittmann (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		We show that P(G; pc, !1) is a polynomial in pc and 11 which simultaneously generalizes the chromatic polynomial, the independence polynomial, and the matching polynomial of a graph.
		The new polynomial satisfies both an edge decomposition formula and a vertex decomposition formula.
		Dohmen and P. Tittmann, A new two-variable generalization of the chromatic polynomial, submitted.
	citeseer.ist.psu.edu /556948.html (240 words)

	Programme
		polynomials of a graph (or binary matroid) and its dual.
		The multivariate Tutte polynomial for graphs and matroids:
		characteristic is the equal to the Jones polynomial of knots.
	www.crm.es /TuttePolynomials/programme.htm (1752 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Introduced by Richard Stanley, the chromatic symmetric function is a generalization of the chromatic polynomial of a graph [1].
		Stanley defines the chromatic symmetric function as a sum of monomial symmetric functions corresponding to proper colorings.
		present a two-variable generalization of the chromatic polynomial, with which they use to verify computationally that there are no non-isomorphic trees of order at most 15 vertices with the same chromatic symmetric function.
	www.cs.wustl.edu /~lt1/csf.html (473 words)

	Citebase - Set maps, umbral calculus, and the chromatic polynomial
		Set maps, umbral calculus, and the chromatic polynomial
		Using umbral calculus, we give a formula for the expansion of such a set map in terms of any polynomial sequence of binomial type.
		This leads to several new expansions of the chromatic polynomial.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0507038 (131 words)

	Atlas: A configuration space approach to the chromatic polynomial, after Eastwood and Huggett by Yongwu Rong (Site not responding. Last check: 2007-11-06)
		This talks is motivated by two pieces of recent work, both realizing the chromatic polynomial of graphs using certain homology groups, but using very different approaches.
		The first is a Khovanov type categorification for the chromatic polynomial, due to Laure Helme-Guizon and the speaker.
		Their construction is natural in the sense that the homology groups of the spaces satisfy a long exact sequence which yileds the well-known deletion-contraction rule.
	atlas-conferences.com /cgi-bin/abstract/capf-14 (227 words)

	ALCOM Seminar (Site not responding. Last check: 2007-11-06)
		We present the first two strongly polynomial algorithms for solving one-player discounted payoff games, running in time O(mn2) and O(mn2 log m), where the latter algorithm allows edges to have different discounting factors.
		We present exact algorithms to compute the chromatic number and the chromatic polynomial in time and space within a polynomial factor of 2
		A brief historical overview is included, including Birkhoff's 1912 definition of the chromatic polynomial.
	www.brics.dk /Activities/ALCOM/index.html (729 words)

	JIPAM - Journal of Inequalities in Pure and Applied Mathematics
		Inequalities of Bonferroni-Galambos Type with Applications to the Tutte Polynomial and the Chromatic Polynomial
		Bonferroni inequalities, Inclusion-exclusion, Tutte polynomial, Chromatic polynomial, Graph, Matroid
		The result is applied to the Tutte polynomial of a matroid and the chromatic polynomial of a graph.
	jipam.vu.edu.au /article.php?sid=423 (69 words)

	Math Forum Discussions
		Could someone please show me how to find or prove the chromatic
		polynomial of a graph with a bridge e.
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	mathforum.org /kb/thread.jspa?threadID=1361834&messageID=4633525 (69 words)

	Math Forum Discussions
		> polynomial of a graph with a bridge e.
		It's not 100% clear what you mean by bridge here, do you mean that
		There is a rule about chromatic polnomials that says that if you can
	mathforum.org /kb/thread.jspa?threadID=1361834&messageID=4633525 (159 words)

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